Hello, about approximations to Pi, we have to take a look at the ratio of the logs (in base 10 to make it simple), on each side, r1 = log of the value of approximation for example Pi = 355/113 is 6.57 decimal digits good, but the size of 355 and 113, have to be considered. If r1 = log(1/(abs(x-a))/log(10), where a is the approximation and x the constant. r2 = r1/(log(max(a(i))*log(10)). then in that case r2 = 2.57. In other words if we have an approximation of Pi for example then the size on each side tells if the approx. is good or not. Another example : 3*log(5280)/sqrt(67) = 9.2 decimal digits and the r2 is 2.474. If r2 is >> 1 then the approximation can be considered good, if r2 is close to 1 : not good, poor or trivial. an example is Pi = 3 + 1/10 + 4/100 + 1/1000 + 5/10000, , the value of r2 is 1 since log(10000)/log(10) = 4. The well known number : log(262537412640768744)/sqrt(163) = Pi, in this case r1 = 30.65 and r2 = 1.754, here a(1) is the max, which is 262537412640768744. An even more precise measure could be, the total size of the coefficients of the approximation, would be called r3 I presume. If r3 = 1, then again, it could be considered as a rewrite of the same information in another base. Reference : http://plouffe.fr/simon/articles/1410.0054v1.pdf best regards, Simon Plouffe Le 2018-02-17 à 02:55, Bill Gosper a écrit :
On 2018-02-16 14:54, françois mendzina essomba2 via math-fun wrote:
Hello
I noticed that:
(9*(34+(3^7)/(10^6)))^(1/5) =3.14159264992426 Log[23 + 25/9 (√8 - 25/9)] = 3.141592653489906 (ries) --rwg
math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun